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As I understand it (this is all from Wikipedia), Turing machines are one of the various possible formal definitions of algorithms (others include recursive functions, lambda-calculus and post-Turing machines). I guess Turing machines is what you have in mind when you say that "[algorithms are] the design plan of a computing machine", i.e. the design plan of a particular Turing machine. Could be, but you are abusing the meaning of "machine" in your next sentence, when you refer to the patentability of machines in general. A computing machine (in Turing's sense) is a thought experiment, not a physical object. That Turing used the word "machine" for his theoretical device does not mean that Turing machines should be considered and treated like pulleys or steam engines. With this in mind, the rest of your reasoning about how benefitial and reasonable patents are is irrelevant to the subject being discussed.

You're basically right with respect to Turing machines, but register machines are similarly flexible, and it's certainly possible to build an actual, physical register machine. I would argue that that's insufficient to permit algorithm patents on its own, though, because the process of transforming an algorithm into a register machine can actually be mostly automated. If you come up with a clever way to do it that's better than any automated approach, sure, that's probably patentable and arguably should be. But the mere fact that such a transformation is possible shouldn't extend patent protection to the algorithm side of it.